D.B.
Was he abusive and controlling when you were together? Did he always get angry when he didn't get his way? That's what this sounds like. He wants the sex, and tried to make it sound like a good decision "for the child"? You were wise to not go that route. Any person who tries to use sex as a bargaining tool so you can earn his "good behavior" is someone you need to steer clear of.
I have a friend with an awful ex. He's always blaming her for everything, including the divorce (even though he moved away and got remarried pretty quickly). And of course he does nothing for the kids except child support (after she took him to court) and no help even though one child has been hospitalized many times. Basically, you can't co-parent with someone like this.
You have 3 choices -
1) Keep him in your life and your child's life for financial reasons or because he wants to do things with her.
2) See if he wants to terminate his parental rights, even though that means you won't have child support and your child will never see her father. He may not go for it anyway since he's so intent on staying in your life and in your face.
3) Do most everything through email and text message so you have a written record of what he says. Keep your side of it "businesslike" and polite, without emotion, so that you can always show the court that you took the high road. This takes incredible patience and it means you have to find someone else (friend, family member, counselor) to unload your stress on when you need to vent. You can also put him in charge of getting info on his own - so he has rights to medical info and can contact the pediatrician on his own, and when your daughter is in preschool or elementary school, he can contact the school to get progress reports and notifications of special events (parent night, etc.). You can send him a copy of her annual physical or any notices from school if you want to, and you can notify him by text that she had a fever of 102 and was taken to the doctor and is now taking XYZ prescription 3 times a day. You have to have a visitation agreement nailed down and filed with the court, and you stick to it to minimize the contact with him to make changes. Try not to let him in your house if he's bossy or pushy or abusive. Get help if he's dangerous. If the "exchange" for visitation is problem-laden, do it at the police station - a lot of divorced parents do that to keep the bad behavior to a minimum. That requirement can even be in the visitation agreement. I think you have to give up on the idea hat he will cooperate, and just hope that he eventually gets tired of fighting you and meets someone else to occupy himself. Part of his fighting is to keep you in his life - but you don't have to play that game if you don't want to.